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In this paper, we study the dynamics of the atomic inversion, scaled atomic Wehrl entropy and marginal atomic Wehrl density for a single two-level atom interacting with SU(1,1) quantum system. We obtain the expectation values of the atomic variables using specific initial conditions. We examine the effects of different parameters on the scaled atomic Wehrl entropy and marginal atomic Wehrl density. We observe an interesting monotonic relation between the different physical quantities for different values of the initial atomic position and detuning parameter.

Quantum entropy, is considered the main generalization of the Boltzmann classical entropy, proposed by von Neumann [

The Wehrl entropy (WE) is more sensitive in distinguishing states than the von Neumann entropy since WE is a state dependent [

Different entanglement measures and quantifiers for mixed and pure states have been proposed, such as the negativity and atomic Wehrl entropy. The relation between mixed state entanglement and the atomic Wehrl entropy (AWE) has not been studied widely. However, there are some attempts to quantify the pure state entanglement by using AWE. In this context, the entanglement evaluation with AWE and atomic Fisher information has been investigated [

Realistic quantum systems are not closed, which causes the rapid destruction of crucial quantum properties. Therefore, the unavoidable interaction between a quantum system, understanding the dynamics of entanglement measures and finding the correlation between different phenomenons may stimulate great interest. In the present article, our main interest is to investigate the evolution of the scaled atomic Wehrl entropy (AWE) of a single two-level atom and SU(1,1) quantum system in the presence of detuning parameter, which leads us to address the question: Can the AWE be used as a indicator of the entanglement and dynamical properties of the system in the presence of non-linear terms?

The article is organized as follows: In Section 2, we introduce the model of the single two-level and SU(1,1) quantum system in the presence of detuning parameter. The definitions of the scaled atomic Wehrl entropy, atomic inversion and marginal atomic Wehrl density are introduced in Section 3. We conclude the main results with some remarks in Section 4.

The Hamiltonian which describe the interaction between a single two-level atom and

where

while

The Heisenberg equation of motion for any operator

thus, the equations of motion for

then

In this case, we have

where

thus

where

where

we note that

also

for simplicity we can write

where

The time evolution for the expectation value of any operator can be calculate through the following relation

Now the initial state of the system can be written as

where

Substituting from Equation (26) in Equation (28), then the final form of the wave function can be written as

Then the wave function can be written in the form

and consequently the density matrix

where

and

Thus the expectation value for any operator can be calculated through

where

Where we have used the abbreviations

In this section, we will use the scaled atomic Wehrl entropy as an entanglement quantifier between single two level atom and SU(1,1) quantum system.

The scaled atomic Wehrl entropy can be written in terms of the atomic Q function as [

(39)

In the above equation

where

where

It is worth noting that from the definition (39) the

By integrating the atomic Wehrl density

In this section, we discuss and present some statistical aspects such as the atomic inversion

The atomic inversion of the atom is one of the important atomic dynamic variables of the system. This in fact would give us information about the behavior of the atom state during interaction time. In

Now we are in a position to discuss the evolution of the marginal atomic Wehrl density

Quantum entanglement is a key resource which distinguishes quantum information theory from classical one. It plays a central role in quantum information and computation. In this paper, we have discussed the problem of the interaction between two-level atom and SU(1,1) quantum system. The model was considered when the twolevel atom is initially in superposition state and the expectation values of the atomic variable are obtained

analytically. Using the scaled atomic Wehrl entropy the system entanglement has been investigated. The analysis herein has been carried out at two distinct considerations of the detuning parameter and initial atomic state setting. Our results show that the SU(1,1) quantum field-atom interaction considering the effect of the initial state setting and detuning parameter has much richer structure. The initial atomic state position and detuning parameter has an important role on the dynamics of the atomic inversion, scaled atomic Wehrl entropy and marginal atomic Wehrl density.